It is well known that \({\mathcal {M}}\)  is bounded on \(L^s(X,d\mu )\)  for all \(1<s<\infty \) ;
see, for instance, [1]}.
The next lemma gives a similar boundedness
result for the local maximal operator \({{\mathcal {M}}}_{\Omega ,\kappa }\) 
on the weighted space \(L^s(\Omega ;w_\beta \,d\mu )\) ,
when \(1<s<\infty \) , \(0<\kappa <1/5\) 
and \(w_\beta (x)=d(x,\Omega ^c)^\beta \) 
is a distance weight, for \(\beta \in \mathbb {R}\)  and \(x\in \Omega \) .
